An analogue of Cobham's theorem for graph directed iterated function systems

Feng and Wang showed that two homogeneous iterated function systems in RR with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend this result to graph directed iterated function systems in RnRn with contraction ratios that are of the form 1β, for integers β. By using a result of Boigelot et al., this allows us to give a proof of a conjecture of Adamczewski and Bell. In doing so, we link the graph directed iterated function systems to Büchi automata. In particular, this link extends to real numbers β  . We introduce a logical formalism that permits to characterize sets of RnRn whose representations in base β are recognized by some Büchi automata. This result depends on the algebraic properties of the base: β being a Pisot or a Parry number. The main motivation of this work is to draw a general picture representing the different frameworks where an analogue of Cobham's theorem is known.